Two rods, one of aluminum and the other made of steel, having initial length $${\ell _1}$$ and $${\ell _2}$$ are connected together to form a single rod of length $${{\ell _1} + {\ell _2}}.$$  The co-efficients of linear expansion for aluminum and steel are $${\alpha _a}$$ and $${\alpha _s}$$ and respectively. If the length of each rod increases by the same amount when their temperature are raised by $${t^ \circ }C,$$  then find the ratio $$\frac{{{\ell _1}}}{{\left( {{\ell _1} + {\ell _2}} \right)}}$$

Solution :
The lengths of each rod increases by the same amount
$$\eqalign{ & \therefore \,\,\Delta {\ell _a} = \Delta {\ell _s} \cr & \Rightarrow \,\,{\ell _1}{\alpha _a}t = {\ell _2}{\alpha _s}t \cr & \Rightarrow \,\,\frac{{{\ell _2}}}{{{\ell _1}}} = \frac{{{\alpha _a}}}{{{\alpha _s}}} \cr & \Rightarrow \,\,\frac{{{\ell _2}}}{{{\ell _1}}} + 1 = \frac{{{\alpha _a}}}{{{\alpha _s}}} + 1 \cr & \Rightarrow \,\,\frac{{{\ell _2} + {\ell _1}}}{{{\ell _1}}} = \frac{{{\alpha _a} + {\alpha _s}}}{{{\alpha _s}}} \cr & \Rightarrow \,\,\frac{{{\ell _1}}}{{{\ell _1} + {\ell _2}}} = \frac{{{\alpha _s}}}{{{\alpha _a} + {\alpha _s}}} \cr} $$